What is a system of linear equations?

A set of two or more linear equations containing the same set of variables.

What is Cramer's Rule?

An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution using determinants.

What happens if the determinant D is zero?

If D = 0, the system has either infinitely many solutions (dependent) or no solution (inconsistent). The calculator will identify this state.

Math

↵ to calculate

Linear Equations Solver — 2 & 3 Variables Solver

Solve systems of linear equations with 2 or 3 variables instantly. Displays step-by-step determinants using Cramer's Rule.

✨

How to use this calculator

👉 Fill in the boxes below and your answer appears instantly — no maths needed, we do it all for you! 🎉

👶

In plain English — what does this do?

🔢 This tool does the maths for you! Just type in your numbers, and it gives you the answer right away. No need to count on your fingers or use a pen and paper.

📌 Direct Answer & Summary

A linear equations solver is a calculator designed to solve systems of linear equations. It finds the values of variables (like x, y, and z) that satisfy multiple equations simultaneously. This calculator supports 2-variable systems (two equations) and 3-variable systems (three equations) and uses Cramer's Rule to solve them, rendering step-by-step determinants.

Equation Format: ax + by = c

Eq 1:

Eq 2:

Calculation Result

x = —

y = —

Determinant Calculations (Cramer's Rule):

D (Main Determinant) = 0

D_x = 0

D_y = 0

What is Linear Equations Solver — 2 & 3 Variables Solver?

How to use it

1️⃣ Select the system type: 2 Variables (2x2) or 3 Variables (3x3).
2️⃣ Enter the coefficients (a, b, c) and constant terms (d) for each equation.
3️⃣ The solutions for x, y, and z update automatically.
4️⃣ Review the determinant details (D, Dx, Dy, Dz) to understand the step-by-step solution.

Formula

For a 2x2 system: a1*x + b1*y = c1, a2*x + b2*y = c2. Determinant D = a1*b2 - a2*b1. Dx = c1*b2 - c2*b1. Dy = a1*c2 - a2*c1. x = Dx/D, y = Dy/D (provided D != 0).

💡 See it in action — a real example

For the system: 2x + 1y = 9, 1x - 1y = 3. Enter a1=2, b1=1, c1=9 and a2=1, b2=-1, c2=3. D = -3, Dx = -12, Dy = -3. x = -12/-3 = 4. y = -3/-3 = 1.

❓ Common questions

What is a system of linear equations?: A set of two or more linear equations containing the same set of variables.
What is Cramer's Rule?: An explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution using determinants.
What happens if the determinant D is zero?: If D = 0, the system has either infinitely many solutions (dependent) or no solution (inconsistent). The calculator will identify this state.

⚡ Related Math Tools & Calculators

Quadratic Equation Calculator Matrix Calculator Scientific Calculator